Multiplication and Division Conceptual Structures for MultiplicationRepeated Addition

This is the first structure that we introduce children to.

It builds on the understanding of addition but in the context of equal sized groups.

Rectangular Array/Area Model

This is often the second representation of multiplication introduced It is useful to show the commutative property that 3 x 4 = 4 x 3 = 12

Number Line A number line can

represent skip counting visually.

Scaling Scaling is the most abstract structure, as it cannot be

understood through counting. Scaling is frequently used in everyday life when comparing

quantities or measuring.

Helpful Manipulatives Base 10 blocks and Graphic Organizer Linking Cubes Counters Plastic or Wooden Squares Mini White Board – grid side Dot Array BLM

Single Digit Multiplication FactsMultiplication facts should be introduced and mastered by relating to existing knowledge. If students are stuck in a ‘counting’ stage – either by ones or skip-counting to know their single-digit multiplication facts, it is important that they understand strategies beyond counting before they practice. Counting is a dangerous stage for students, as they can get stuck in this inefficient and often inaccurate stage. Students should not move to multi-digit multiplication before they understand multiplication strategies for single-digit multiplication.

It is important that students understand the commutative property 2 x 4 = 8 and 4 x 2 = 8. 2 x 4 should be related to the addition fact 4 + 4 = 8, or double 4. Using a multiplication table as a visual structure is helpful to see patterns in multiplication facts.

Mental Strategies Continuum Same as (1 facts) Doubles. (2 facts) Doubles and 1 more (3 facts) Double Doubles (4 facts)

Tens and fives (10, 5 facts) Relating to tens (9 facts) Remaining facts (6, 7, 8 facts)

Conceptual Structures for DivisionEqual Grouping In an equal grouping (quotition) question, the total number are known, and the size of each group is known.

• The unknown is how many groups there are.

Equal SharingIn an equal sharing (partition) question, the total number are known, and the number of groups is known.

• The unknown is how many are in each group.

Number Line

RatioThis is a comparison of the scale of two quantities and is often referred to as scale factor. This is a difficult concept as you can’t subtract to find the ratio.

Division FactsRelate division facts back to multiplication facts families:

Ex) 6 x 8 = 48

8 x 6 = 48

48 ÷ 6 = 8

48 ÷ 8 = 6

Multi-Digit Multiplication and DivisionConcrete

Concrete base 10 blocks can be used to represent up to 2 digit x 2 digit multiplication or up to 4 digit ÷ 2 digit division.

Multiplication: Using Base 10 Blocks. Watch the video that explains how to use a Base 10 Block Organizer and blocks to understand multi-digit multiplication.

Division: You can use the same organizer to understand division. o In the top band, put the divisor given. o Select the number of base 10 blocks that represent the dividend. These go into the area. o Remind students that the blocks can be exchanged. For example, 10 x ten rods = 100

block.o Try to make a rectangle with one side length equal to the divisor. Students may need to

exchange rods to make a perfect rectangle.o How long is the other side of the rectangle? This is the quotient.

Pictorial The closest model to concrete is the rectangle section method. It uses grid

paper to create a scale area model representation. It is important that students know how to decompose numbers.

Similar to the rectangle section method is the area, or box model. This is not drawn to scale, but preserves understanding of place value. It is also helpful as a step towards working with polynomials in later grades and is helpful when building a conceptual understanding of multiplication and division.

Multiplication: o Decompose the two numbers being multiplied.o Label the two sides of the rectangle.o Multiply each row and column, then add those products together.

Division: o Decompose the divisor. This divisor is used to label one side of the

rectangle. The dividend is put into the area of the rectangle. Watch the video to see an example of the box method.

Symbolic The partial product and partial quotient method also use decomposition of numbers to multiply

and divide numbers. It is similar in thinking to the area model, but does not organize the work into a rectangle.

The Gelosia method of multiplication is very efficient. It can be applied to decimal numbers and multi-digit numbers. Similar to the traditional algorithm, it does not preserve an understanding of place value, so should not be introduced until students have a conceptual understanding of multiplication.

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